On idempotent discrete uninorms

Miguel Couceiro, Jimmy Devillet, and Jean-Luc Marichal

Abstract In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conser- vative if it always outputs one of its input values. More precisely we first show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing in each variable. Then we show that, in this char- acterization, symmetry can be replaced with both bisymmetry and existence of a neutral element.

1 Introduction

Aggregation functions defined on linguistic scales (i.e., finite chains) have been in- tensively investigated for about two decades; see, e.g., [1–4, 6–11, 13, 14]. Among these functions, discrete fuzzy connectives (such as discrete uninorms) are associa- tive binary operations that play an important role in fuzzy logic.

This short paper focuses on characterizations of the class of idempotent discrete uninorms. Recall that a discrete uninorm is a binary operation on a finite chain that is associative, symmetric, nondecreasing in each variable, and has a neutral element.

A first characterization of the class of idempotent discrete uninorms was given by De Baets et al. [1, Theorem 3]. This characterization reveals that any idempotent

Miguel Couceiro

LORIA, (CNRS - Inria Nancy Grand Est - Universit´e de Lorraine), BP239, 54506 Vandoeuvre-l`es- Nancy, France, e-mail: [email protected]

Jimmy Devillet

Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg e-mail: [email protected]

Jean-Luc Marichal

Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg e-mail: [email protected]

1

discrete uninorm is a combination of the minimum and maximum operations. In particular, such an operation is conservative in the sense that it always outputs one of the input values.

The outline of this paper is as follows. After presenting some preliminary results on conservative operations in Section 2, we show in Section 3 that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing in each variable. This new characterization is very simple and requires neither associativity nor the existence of a neutral element. In Section 4 we provide an alternative characterization of this class in terms of the bisymmetry prop- erty. More specifically, we show that the idempotent discrete uninorms are exactly those operations that are conservative, bisymmetric, nondecreasing in each variable, and have neutral elements.

2 Preliminaries

In this section we present some basic definitions and preliminary results.

Let X be an arbitrary nonempty set and let ∆

= { (x, x) | x ∈ X } . Definition 1. An operation F: X

→ X is said to be

• idempotent if F (x, x) = x for all x ∈ X .

• conservative if F (x, y) ∈ { x, y } for all x, y ∈ X .

• associative if F(F (x, y), z) = F(x, F(y,z)) for all x, y, z ∈ X .

An element e ∈ X is said to be a neutral element of F (or simply a neutral element) if F(x,e) = F(e,x) = x for all x ∈ X. In this case we easily show by contradiction that such a neutral element is unique. The points (x, y) and (u, v) of X

are said to be connected for F (or simply connected) if F(x, y) = F (u, v). We observe that “being connected” is an equivalence relation. The point (x, y) of X

is said to be isolated for F (or simply isolated) if it is not connected to another point in X

.

Remark 1. Conservativeness was introduced in Pouzet et al. [12]. This condition is also called “local internality” in Mart´ın et al. [5].

Lemma 1. Let F: X

→ X be an idempotent operation. If the point (x, y) ∈ X

is isolated, then it lies on ∆

, that is, x = y.

Remark 2. We observe that idempotency is necessary in Lemma 1. Indeed, consider the operation F: X

→ X, where X = { a, b } , defined as F (x, y) = a, if (x,y) = (a, b), and F(x,y) = b, otherwise. Then (a, b) is isolated and a ̸ = b. The contour plot of F is represented in Figure 1. Here and throughout, connected points are joined by edges.

To keep the figures simple we sometimes omit the edges obtained by transitivity.

The following lemma provides an easy test for the existence of a neutral element

of a conservative operation.

s s

s s

(a,a) (b,a)

(a,b) (b,b)

Fig. 1 A non-idempotent operation

Lemma 2. Let F: X

→ X be a conservative operation and let e ∈ X . Then e is a neutral element if and only if (e, e) is isolated.

Corollary 1. Any isolated point (x, y) of a conservative operation F: X

→ X is unique and lies on ∆

. Moreover, x = y is a neutral element.

Remark 3. Lemma 2 no longer holds if conservativeness is relaxed into idempo- tency. Indeed, by simply taking X = { a, b, c } we can easily construct an idempotent operation with an isolated point on ∆

and no neutral element (see Figure 2). Also, it is easy to construct an idempotent operation with a neutral element and no isolated point (see Figure 3). It is also noteworthy that there are idempotent operations with more than one isolated point (see Figure 4).

s s s

s s s

s s s

(a,a) (b,a) (c,a)

(a,b) (b,b) (c,b)

(a,c) (b,c) (c,c)

Fig. 2 An operation with no neutral element

3 Main results

We now focus on characterizations of the class of idempotent discrete uninorms.

These operations are defined on finite chains. Without loss of generality we will

only consider the n-element chains L

= { 1, . . . , n } , n ≥ 1, endowed with the usual

ordering relation ≤ .

s s s

s s s

s s s

(a,a) (b,a) (c,a)

(a,b) (b,b) (c,b)

(a,c) (b,c) (c,c)

@ @

@ @

@ @

@ @

Fig. 3 An operation with no isolated point

s s s

s s s

s s s

(a,a) (b,a) (c,a)

(a,b) (b,b) (c,b)

(a,c) (b,c) (c,c)

Fig. 4 An operation with two isolated points

Recall that an operation F :L

→ L

is said to be nondecreasing in each variable if F(x, y) ≤ F (x

, y

) whenever x ≤ x

and y ≤ y

.

Definition 2 (see, e.g., [1]). A discrete uninorm on L

is an operation U: L

→ L

that is associative, symmetric, nondecreasing in each variable, and has a neutral element.

A characterization of the class of idempotent discrete uninorms is given in the following theorem. Although this characterization is somewhat intricate, it shows, together with Lemma 3 below, that any idempotent discrete uninorm is conservative.

Theorem 1 (see [1, Theorem 3]). An operation F: L

→ L

with a neutral element 1 < e < n is an idempotent discrete uninorm if and only if there exists a nonincreas- ing map g: [1,e] → [e, n] (nonincreasing means that g(x) ≥ g(y) whenever x ≤ y), with g(e) = e, such that

F(x, y) =

{ min { x, y } , if y ≤ g(x) and x ≤ g(1), max { x, y } , otherwise,

where g: L

→ L

is defined by g(x) =

 



g(x), if x ≤ e,

max { z ∈ [1, e] | g(z) ≥ x } , if e ≤ x ≤ g(1),

1, if x > g(1).

We now show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing in each variable (see Theo- rem 2).

First consider the following lemma, which actually holds on arbitrary, not neces- sarily finite, chains.

Lemma 3. If F: L

→ L

is idempotent, nondecreasing in each variable, and has a neutral element e ∈ L

, then F |

= min and F |

= max.

Proposition 1. If F : L

→ L

is conservative, symmetric, and nondecreasing in each

variable, then it is associative and it has a neutral element.

For n = 2 and n = 3, the possible operations F: L

→ L

that are conservative, symmetric, and nondecreasing in each variable have contour plots depicted in Fig- ures 5 and 6, respectively.

s s

s s

(1,1) (2,1)

(1,2) (2,2)

s s

s s

(1,1) (2,1)

(1,2) (2,2)

Fig. 5 Possible operations whenn=2

s s s

s s s

s s s

(1,1) (2,1) (3,1)

(1,2) (2,2) (3,2)

(1,3) (2,3) (3,3)

s s s

s s s

s s s

(1,1) (2,1) (3,1)

(1,2) (2,2) (3,2)

(1,3) (2,3) (3,3)

s s s

s s s

s s s

(1,1) (2,1) (3,1)

(1,2) (2,2) (3,2)

(1,3) (2,3) (3,3)

s s s

s s s

s s s

(1,1) (2,1) (3,1)

(1,2) (2,2) (3,2)

(1,3) (2,3) (3,3)

Fig. 6 Possible operations whenn=3

Remark 4. (a) The existence of a neutral element in Proposition 1 is no longer guar- anteed if the chain is not finite. For instance, the real operation F: [0, 1]

→ [0, 1]

defined by F(x, y) = min { x, y } , if x,y ∈ [0,

)

, and F (x, y) = max { x, y } , other-

wise, is conservative, symmetric, and nondecreasing in each variable, but it does

not have a neutral element.

(b) We observe that conservativeness cannot be relaxed into idempotency in Propo- sition 1. For instance the operation F: L

→ L

whose contour plot is depicted in Figure 2 is idempotent, symmetric, and nondecreasing in each variable, but one can show that it is not associative and it has no neutral element.

(c) We also observe that each of the conditions of Proposition 1 is necessary. Indeed, we give in Figure 7 an operation that is conservative and symmetric but that is not nondecreasing in each variable. We also give in Figure 8 an operation that is conservative and nondecreasing in each variable but not symmetric. Finally, we give in Figure 9 an operation that is symmetric and nondecreasing in each variable but not conservative. None of these three operations is associative and none has a neutral element.

s s s

s s s

s s s

2

3

Fig. 7 An operation that fails to be nondecreasing in each variable

s s s

s s s

s s s

1

2

3

Fig. 8 An operation that fails to be symmetric

s s s

s s s

s s s

1

3

Fig. 9 An operation that fails to be conservative

Theorem 2. An operation F: L

→ L

is conservative, symmetric, and nondecreas- ing in each variable if and only if it is an idempotent discrete uninorm. Moreover, there are exactly 2

such operations.

Remark 5. Theorem 2 enables us to provide a graphical characterization of the idem-

potent discrete uninorms in terms of their contour plots. Indeed, denoting by L an

arbitrary n-element chain, we observe that the restriction F |

of any idempotent discrete uninorm F: L

→ L to any subchain L

obtained by removing one of the endpoints of L is also an idempotent discrete uninorm. Moreover, the operation F (or equivalently its contour plot) can be retrieved from F |

by connecting all the points of L

\ L

. It follows that all the idempotent discrete uninorms can be con- structed recursively in terms of their contour plots.

4 Bisymmetric operations

In this section we provide a characterization of the class of idempotent discrete uninorms in terms of the bisymmetry (or mediality) property.

Definition 3. An operation F: X

→ X is said to be bisymmetric if F(F(x, y), F(u, v)) = F (F (x, u), F(y, v)) for all x,y, u,v ∈ X.

Proposition 2. Let F: X

→ X be a conservative operation that has a neutral ele- ment. Then F is bisymmetric if and only if it is associative and symmetric.

Combining Proposition 2 with Theorem 2, we can easily derive the following alternative characterization of idempotent discrete uninorms.

Theorem 3. An operation F :L

→ L

is conservative, bisymmetric, nondecreasing in each variable, and has a neutral element if and only if it is an idempotent discrete uninorm.

Acknowledgements The authors thank Gergely Kiss for fruitful discussion and valuable remarks.

This research is partly supported by the internal research project R-AGR-0500-MRO3 of the Uni- versity of Luxembourg.

References

1. De Baets, B., Fodor, J., Ruiz-Aguilera, D., Torrens, J.: Idempotent uninorms on finite ordinal scales. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems17(1),1–14 (2009) 2. De Baets, B., Mesiar, R.: Discrete triangular norms. In: Rodabaugh, S., Klement, E. P. (eds.)

Topological and Algebraic Structures in Fuzzy Sets, A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic, pp. 389–400. Kluwer Academic Publishers (2003)

3. Fodor, J.: Smooth associative operations on finite ordinal scales. IEEE Trans. Fuzzy Systems 8, 791–795 (2000)

4. Li, G., Liu, H.-W., Fodor, J.: On weakly smooth uninorms on finite chain. Int. J. Intelligent Systems30, 421–440 (2015)

5. Mart´ın, J., Mayor, G., Torrens, J.: On locally internal monotonic operations Fuzzy Sets and Systems137, 27-42 (2003)

6. Mas, M., Mayor, G., Torrens, J.: t-operators and uninorms on a finite totally ordered set. Int.

J. Intelligent Systems14, 909–922 (1999)

7. Mas, M., Monserrat, M., Torrens, J.: On bisymmetric operators on a finite chain. IEEE Trans.

Fuzzy Systems11, 647–651 (2003)

8. Mas, M., Monserrat, M., and Torrens, J.: On left and right uninorms on a finite chain. Fuzzy Sets and Systems146, 3–17 (2004)

9. Mas, M., Monserrat, M., Torrens, J.: Smooth t-subnorms on finite scales. Fuzzy Sets and Systems167, 82-91 (2011)

10. Mayor, G., Su˜ner, J., Torrens, J.: Copula-like operations on finite settings. IEEE Trans. Fuzzy Systems13, 468-477 (2005)

11. Mayor, G., Torrens, J.: Triangular norms in discrete settings. In: Klement, E. P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp.

189–230. Elsevier, Amsterdam (2005)

12. Pouzet, M., Rosenberg, I. G., Stone, M. G.: A projection property. Algebra Universalis36(2), 159–184 (1996)

13. Ruiz-Aguilera, D., Torrens, J.: A characterization of discrete uninorms having smooth under- lying operators. Fuzzy Sets and Systems268, 44-58 (2015)

14. Su, Y., Liu, H.-W.: Discrete aggregation operators with annihilator. Fuzzy Sets and Systems 308, 72-84 (2017)